where \(a_{1}\)  (\(a_{2}\) ) is the magnitude of the spin vector of the primary (secondary) BH, and \(\alpha _\chi \)  and \(\beta _\chi \)  are population hyperparameters determining the structure of the Beta distribution. The model for the distribution of tilt angles, \(\theta \) , is motivated by contribution from two subpopulations: one preferentially aligned (\(\cos (\theta ) \approx 1\) ) and one isotropic [1]}, [2]}, [3]}. The model is parameterized as:
\(\pi (\cos \theta _{1,2}|\xi , \sigma _t) = \xi G_{t}(\cos \theta _1|\sigma _t)G_{t}(\cos \theta _2|\sigma _t) + \frac{1 - \xi }{4},\) 
